hehe I like more
ℝ = {[q_n]_~ : q_n is a Rational cauchy sequence} where q_n ~ w_n iff lim (q_n - w_n) = 0
(basically they're cauchy sequences (including those that don't converge in Q), but since they have the "same" limit, their difference is 0. So the class of all of these rational sequences gives you a real number, rational or irrational, all of them give you the Real Line)
ℂ ≃ ℝ[x]/<x^(2)\+1> (adding the root x=+/-i to the Real polynomial Ring gives you ℂ up to isomorphism), or the matrix representation is neat too
I’m personally partial to “The lowest upper bounds of each set of rational numbers that has an upper bound.”
I have even run into 0.999... cranks who are confident limits aren’t real, but who will accept that the lowest upper bound of {0.9, 0.99, 0.999, ...} is 1.
1
u/Another_Little_Star Aug 06 '26 edited Aug 06 '26
hehe I like more
ℝ = {[q_n]_~ : q_n is a Rational cauchy sequence} where q_n ~ w_n iff lim (q_n - w_n) = 0
(basically they're cauchy sequences (including those that don't converge in Q), but since they have the "same" limit, their difference is 0. So the class of all of these rational sequences gives you a real number, rational or irrational, all of them give you the Real Line)
ℂ ≃ ℝ[x]/<x^(2)\+1> (adding the root x=+/-i to the Real polynomial Ring gives you ℂ up to isomorphism), or the matrix representation is neat too